Seven ways to visualize functions

I describe how mathematical functions can be visualized by:
(1) elements and arrows
(2) Wiring diagrams
(3) fibres
(4) bouquets
(5) graphs
(6) ontology logs
(7) categories
These ideas have applications throughout mathematics and beyond. Notable applications include set theory, cooking, planning, functional programming, dependent type theory, linguistics and category theory.
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[@spirit6221]

Excellent

[@SliversRebuilt]

"Obviously no one owns -1 cars"
The repo man would beg to differ ; )
(Btw this is one of the best videos on the internet, like many of yours are! You're the man, dude, thanks so much for all of this you've shared with us over the years 🙏🏻🤟🏻)

[@bernardofitzpatrick5403]

Great intro to functions and category theory. Will be looking at ure series on category theory next 🤙🏽

[@Leonar1450]

I love these videos about how you can view something, even that you feel very familiar with, in so many different ways to better suit different scenarios, situations of even people's ways of thinking. Love it!

[@hansenmarc]

Fascinating! I’d never seen the fiber and petal representations. Thank you for sharing.

[@tantzer6113]

Wiring diagrams are common in linear systems theory. Each box represents a linear system (representable by differential equations) that operates on a “signal,” i.e. a time series (discrete) or continuous function of time. The equivalent of Lab-view for signals and systems would be Matlab’s Simulink.

[@devon9374]

Just bought your book! Can’t wait for it to arrive :)

[@TheLogvasStudio]

More AMSR like at the beginning of this video, please.

[@Kirshach]

Thank you so much for what you do, Richard! Your work is invaluable

[@probablyshadman]

Love from Bangladesh 🖤🖤🇧🇩

[@Marzaries]

This video inspired me to think of a metaphor which helped me have a more distinct understanding of a function in my linear algebra studies (be warned, I'm a complete beginner).
Functions can be denoted by a pair things and their relationships. one then looks for a relationship in this pair, and this can happen in a variety of ways. For example, I could have the topic of (Fitness), a physical activity carried out consistently over time. But what does this mean?
So the pair of things in discussion could be the relationship between exercise in discrete time, and exercise in continuous time. Let exercise in isolated time be (P) and exercise in continuous time be (Q) and lastly let Y = (P) * (Q). Here Y is the constant and P and Q are variables which are the subjects to change (Y). Whatever variation is shown within P and Q will lead us to concluding whatever relationship they have to (Y). In this case (Y) for the variables (P and Q).

[@nihil45]

Sir, you're an angel.

[@muttdev]

Thanks for the Video Richard. Many times you mentioned that "functions are used to compare objects". Could you please explain how functions are used to compare objects with an example. I understand the concept that a function can be used to map one object into another, but I am not able to understand how it is used to compare one object with another.

[@RichardSouthwell]

For example, a function f from A to B is called a surjection if each member of B has some member of A sent to it with f. If there is a surjection from A to B then A has at least as many members as B. The presence of different kinds of functions tells you how different sets are related.

[@bibliusz777]

have u seen the software called Dreams?

[@RichardSouthwell]

A few peices of software have that name. Could you give a link to be more specific ?

[@bibliusz777]

​@@RichardSouthwell ruclips.net/video/oEl3Ce5MPVg/видео.html
it's really good at visualization and developing in it is really smooth
i think it's the best standard one can get from an editor these days and i'd like to make such editor for programming with rich type theory for specification and optimization purposes

[@nneisler]

Would you say that the function is the basis of modern mathematics?

[@RichardSouthwell]

They sure are important. So are sets. I'd say functions are also quite fundamental to computer science

[@trongtue8384]

I guess your spatial reasoning is very high than your non-verbal